--
--
-- Equations
--
--

def solve eqs := solve' eqs []
  where
    solve1 f expr x := inverse expr f x

    solve' eqs rets :=
      match eqs as list (mathExpr, mathExpr, symbolExpr) with
        | [] -> rets
        | ($f, $expr, $x) :: $rs ->
          solve'
            rs
            (rets ++ [(x, solve1 (substitute rets f) (substitute rets expr) x)])

--
-- Quadratic Equations
--
def quadraticFormula := qF

def qF f x :=
  match coefficients f x as list mathExpr with
    | $a_0 :: $a_1 :: $a_2 :: [] -> qF' a_2 a_1 a_0

def qF' a b c :=
  ( ((- b) + sqrt (b ^ 2 - 4 * a * c)) / 2 * a
  , ((- b) - sqrt (b ^ 2 - 4 * a * c)) / 2 * a )

--
-- Cubic Equations
--
def cubicFormula := cF

def cF f x :=
  match coefficients f x as list mathExpr with
    | $a_0 :: $a_1 :: $a_2 :: $a_3 :: [] -> cF' a_3 a_2 a_1 a_0

def cF' a b c d :=
  match (a, b, c, d) as (mathExpr, mathExpr, mathExpr, mathExpr) with
    | (#1, #0, $p, $q) ->
      let (s1, s2) := (\(x, y) -> (rt 3 x, rt 3 y)) (qF' 1 (27 * q) ((-27) * p ^ 3))
       in ( (s1 + s2) / 3               -- r1
          , (w ^ 2 * s1 + w * s2) / 3   -- r2
          , (w * s1 + w ^ 2 * s2) / 3)  -- r3
    | (#1, _, _, _) ->
      let (s1, s2, s3) := withSymbols [x, y]
          cF (substitute [(x, y - b / 3)] (x ^ 3 + b * x ^ 2 + c * x + d)) y
       in (s1 - b / 3, s2 - b / 3, s3 - b / 3)
    | (_, _, _, _) -> cF' 1 (b / a) (c / a) (d / a)